Introduced by R. Schwartz 16 years ago, the pentagram map acts on non-degenerate plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. I shall demonstrate that the pentagram map has an invariant Poisson structure and is completely integrable. I shall also explain that the pentagram map is a discretization of the Boussinesq equation, a well known completely integrable system.
Based on a joint work with V. Ovsienko and R. Schwartz.